Regularity for axisymmetric Navier-Stokes with an Euler length

@misc{constantin2026regularity,
  title = {{Regularity for axisymmetric Navier-Stokes with an Euler length}},
  author = {Peter Constantin and Mihaela Ignatova and Vlad Vicol},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20762},
  eprint = {2609.20762},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20762},
}

TY  - UNPB
TI  - Regularity for axisymmetric Navier-Stokes with an Euler length
AU  - Peter Constantin
AU  - Mihaela Ignatova
AU  - Vlad Vicol
PY  - 2026
DA  - 2026/09/29
PB  - arXiv
DO  - 10.48550/arxiv.2609.20762
UR  - https://arxiv.org/abs/2609.20762
AB  - We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time t=0, and satisfy Type~II pointwise bounds at a vanishing length scale ℓ(t). We say ℓ(t) is an Euler length if it is non-increasing, satisfies a doubling condition, and if ℓ(t)→0 and (−t)/ℓ(t)^2→0 as t→0^-. This includes power laws ℓ(t)=(−t)^γ with 0<γ<1/2, and logarithmic enlargements of the parabolic length. Our main result shows that local bounds of the type |u(·,t)| ≤ Cℓ(t)/(−t) and |∇^2 u(·,t)|≤ C/((−t)ℓ(t)) for all t∈ (−1,0) imply regularity. The proof adapts the circulation and potential-vorticity argument of our earlier paper~\cite{CIV26} to ancient limits obtained from the rescaled vorticity system by zooming in. We show that the second derivative a priori assumption may be replaced by a Hölder bound on the azimuthal vorticity and a corresponding bound on its potential vorticity.
ER  -